nLab
regular differential operator in noncommutative geometry

Regular differential operators have been nontrivially generalized to noncommutative rings (and schemes) by V. Lunts and A. L. Rosenberg, as well as to the setting of braided monoidal categories. As in the commutative case, regular differential operators on a kk-algebra RR form the differential part of the bimodule of the kk-endomorphisms. The differential part is geometrically defined as the Δ\Delta-part where Δ\Delta is the so-called diagonal topologizing subcategory of the abelian category of endofunctors EndcAEnd_c A of the category AA of quasicoherent sheaves (RR-modules in affine case) having right adjoint. The diagonal is by the definition the smallest coreflective topologizing subcategory in EndcAEnd_c A containing the identity functor. For every coreflective topologizing subcategory 𝕋\mathbb{T} in the abelian category satisfying the property sup one defines the notions of 𝕋\mathbb{T}-torsion and 𝕋\mathbb{T}-part of any object MM, see differential monad.

The following two papers dwell mainly on the affine and projective cases

Their motivation is an analogue of a Beilinson-Bernstein localization theorem for quantum groups. The category of differential bimodules is categorically characterized in their work as the minimal coreflective topologizingmonoidal subcategory of the abelian monoidal category of RR-RR-bimodules which is containing RR. In the case of noncommutative rings, Lunts-Rosenberg definition of differential operators has been recovered from a different perspective in the setup of noncommutative algebraic geometry represented by monoidal categories; the emphasis is on the duality between infinitesimals and differential operators:

There are some other approaches to rings of differential operators in noncommutative geometry. In easy semicommutative cases (like nilpotent thickenings of commutative schemes) one can use the standard Grothendieck definition without change. On the other hand, there is an approahc by generators and relations in affine case, corresponding to the recipe for preprojective algebras of quivers. It has some nice localization properties and relations to double derivations and double Poisson geometry. See papers by Yuri Berest and